Matrices

Let $A=\left(\begin{matrix} 1+i & 1 \\ -i & 0 \end{matrix}\right)$where $i=\sqrt{-1}$. Then, the number of elements in the set $\left{n\in\,{1,2,\ldots\,.,100}:A^{n}=A\right}$is

Let$S=\left{n\in\,N,\left(\begin{matrix} 0 & i \\ 1 & 0 \end{matrix}\right)^{n}\left(\begin{matrix} a & b \\ c & d \end{matrix}\right)=\left(\begin{matrix} a & b \\ c & d…

Let $S=${$\sqrt{n}:1⩽n⩽50$and $n$is odd}.Let $a\in\,S$and $A=\left[\begin{matrix} 1 & 0 & a \\ -1 & 1 & 0 \\ -a & 0 & 1 \end{matrix}\right]$.If…

Let $S$be the set containing all $3\times\,3$matrices with entries from $\left{-1,0,1\right}$. The total number of matrices $A\in\,S$such that the sum of all the diagonal elements of $A^{T}A$is $6$is ______.

Let $A=\left[\begin{matrix} \begin{matrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{matrix} \end{matrix}\right]$ and $B=7A^{20}-20A^{7}+2I$, where $I$ is an identity matrix of order $3\times\,3.$ If…

Let $A=\left[\begin{matrix} x & y & z \\ y & z & x \\ z & x & y \end{matrix}\right],$where $x,y$ and $z$ are real numbers such that $x+y+z>0$and $xyz=2$. If $A^{2}=I_{3}$, then the value of $x^{3}+y^{3}+z^{3}$ is

If $A=\left[\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}\right]$ and $M=A+A^{2}+A^{3}+\ldots\,+A^{20},$then the sum of all the elements of the matrix $M$ is equal to _______.

Let$\theta\,=\frac{\pi\,}{5}$and$A=\left[\begin{matrix} \mathrm{cos\theta} & \mathrm{sin\theta} \\ -\mathrm{sin\theta} & \mathrm{cos\theta} \end{matrix}\right]$. If$B=A+A^{4}$, then det$\left(B\right)$:

Let $A=\left[\begin{matrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{matrix}\right]$ and $B=A-I$. If $\omega\,=\frac{\sqrt{3}i-1}{2}$, then the number of elements in the set…

Let $A=\left(\begin{matrix} 2 & -2 \\ 1 & -1 \end{matrix}\right)$and$B=\left(\begin{matrix} -1 & 2 \\ -1 & 2 \end{matrix}\right)$ . Then the number of elements in the set…

Let $A=\left[\begin{matrix} 2 & 3 \\ a & 0 \end{matrix}\right],a\in\,R$ be written as $P+Q$ where $P$is a symmetric matrix and $Q$ is skew symmetric matrix. If $\operatorname{det}\,\left(Q\right)=9$, then the modulus of…

Let$M=\left[\begin{matrix} 0 & -\alpha\, \\ \alpha\, & 0 \end{matrix}\right]$,where $\alpha\,$ is a non-zero realnumber and$N=\underset{k=1}{\overset{49}{\sum\,}}M^{2k}$. If$\left(I-M^{2}\right)N=-2I$, thenthe positive…